Using Feynman’s representation, the equilibrium state of the Bose gas can be described in terms of a probabilistic model, similar to models from non-quantum statistical mechanics. In this model a particle configuration is characterized by the positions of the particles in combination with a permutation of the particles. The natural mathematical
framework for such a particle model is an infinite volume description, but in the case of
the Bose gas model this infinite volume description is still missing: The main difficulty here is to incorporate the particle permutations into the model.

In order to show how to overcome this difficulty we consider a toy model, where the particle positions are fixed at the vertices of the integer lattice. We introduce an infinite volume description of this model in terms of Gibbs measures on permutations of
the integers. This model can then be analyzed by means of geometric tools (based on
a quantity called the flux of a permutation) in combination with probability estimates
and cut-block arguments.

Our result provides a full classification of possible states of the system: For every temperature there are infinitely many possible states, all of them are translation invariant, and exactly one of them is concentrated on permutations consisting of finite cycles only.